Product DescriptionThird Edition: This text book serves as an introduction to logic and set theory. This book is targeted toward non-science students and prospective elementary school teachers who seek to improve their skills in logical thinking and organization of information. ... Read more

Product DescriptionLucidly and gradually explains sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and first-order theories. Its clarity makes this book excellent for self-study. ... Read more

Customer Reviews (5)

Good for self-study or nonmajors
It is really more about foundational issues than sets and logic. The preface says this was intended as a one-year course in the foundations of math for upper division math majors. The delivery is slow and gentle, rather wordy, and a bit stodgy -- not always crystal clear about what point he is making. It is suitable for students who have no experience with higher math. I don't know about students at the author's school, but I think it would try the patience of most seniors or grad students in math. I would recommend it more to lower division and philosophy majors.

Very Terse Treatment of a Broad Range of Topics
If you are already familiar with the material, this book is a concise and clear reference, and yes a great buy. But for learning these topics from the beginning, you would be better served by other books that are focused on just a particular topic.

For example, for logic in the context of set theory, I highly recommend Daniel Velleman's How to Prove it.

Incredible Best Buy
This book is without peer in its breadth of coverage of the foundations of mathematics and logic. I have given this book only 4 stars, because its treatment of any given topicis not classic. It is the total package that astounds.For a mere $15, you get a challenging undergraduate introduction to all of the following topics. I have written in parentheses the names of authors of more definitive treatments:

Stoll's style is quite discursive, far from the terse lemma-theorem-corollary-remark style of so much 20th century mathematics. My only major disappointment is that the formal proof technique set out in chpt. 4 is natural deduction rather than the tableau method or Quine's Main Method.

It is indeed the case that there are no solutions to the exercises, but I do not believe that that is a major flaw.

Unusually clear treatment of very abstract matters.
This book is a great bargain: intuitive and axiomatic set theory, foundations of number systems, first order logic and its completeness and undecidability, the basics of abstract algebra, especially Boolean algebra (through the Stone theorem), elementary group theory, and Godelian incompletability. All in one inexpensive paperback. Excellent coverage of the three way crossroads where logic, modern algebra, and metamathematics intersect. Often the first reference I consult on basic logic.

Even though I am not a mathematician, I can understand, with effort, most of what the author is trying to say.

Good book that falls short of being a great book.
This book is very well written and easy to understand.However, it has a very serious shortcoming: there are no solutions to the exercises.If you're looking for a basic reference, this book is good, but if you want a book you can use to learn set theory and logic, get one that has solutions to the exercises.
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Product DescriptionIn this introduction to set theory and logic, the author discusses first order logic, and gives a rigorous axiomatic presentation of Zermelo-Fraenkel set theory. He includes many methodological remarks and explanations, and demonstrates how the basic concepts of mathematics can be reduced to set theory. He explains concepts and results of recursion theory in intuitive terms, and reaches the limitative results of Skolem, Tarski, Church and Gödel (the celebrated incompleteness theorems). For students of mathematics and philosophy, this book provides an excellent introduction to logic and set theory. ... Read more

Customer Reviews (1)

An excellent introduction to mathematical logic
This is an outstanding undergraduate treatment of the following topics at the core of modern mathematics: set theory, equivalence and ordering relations, cardinals, ordinals, sentential and quantifier logic, and just enough recursion theory to derive the paradoxical undecidability theorems.

The treatment is thoroughly contemporary (eg, Hintikka sets) but not too difficult, because this text emerged out of the philosophy rather than the mathematics classroom. The text is written in the mathematical tradition, consisting of many terse numbered subsections, each containing a definition, theorem, remark, or problem. The organisation of the subject, and the index, are excellent. The author chooses an unusual pedagogic strategy: to present a bit of axiomatic set theory BEFORE the chapters on logic and metamathematics. A benefit of this choice is that the metalanguage employed to exposit these topics can draw freely on elementary set theoretic concepts. This strategy is, of course, a consequence of the triumph of model theory over proof theory.

Except for the omission of Boolean algebra, this is the finest contemporary treatment I know of the propositional calculus.

The book ends with a concise and elegant treatment of the classic limitative results inherent in any formalization of arithmetic. Tarski indefinability is proved quite early on, right after the proof of the diagonal lemma and before the axiomatic systems of arithmetic are set out in order of increasing strength, RR=>Q=>P. The famous theorem of Matyasevich (recursively enumerable relations are diophantine) is stated without proof, then invoked to make stronger and simpler the proof of Godelian incompleteness.

This book is filled with the sort of shrewd philosophical asides too seldom included in textbooks, and that when spoken are the hallmark of a good experienced teacher.
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This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before.

Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted.

Masterwork
"Set Theory and It's Logic" has cast a long shadow since it's first appearance. It is quite simply the crowning work of one of the 20th century's most distinguished philosophers. Sober, clear, and direct (and yet unpretentious and very friendly), it is illuminating to anyone who has the patience to slowly sip it and consider the the way the arguments build up from line to line and page to page a seemingly indestructible house of cards. If one is unprepared for the rigors of the book, that can be easily remedied--Quine also wrote the best introductory book on Logic, "Methods of Logic", which he took through several editions before his death. That book assumes no backround in logic, and a beginner who works her way through the exercises will find herself well-prepared for the magic tricks in "Set Theory and its Logic". I wished we taught this stuff in the public schools, along with mathematics, and (to keep the old dialectic rolling) Homer, Shakespeare, Dante, Tolstoy, Dickinson--the great poets. I might be dreaming, but we may have, at the very least, more ethical scientists, more humane poets, or just plain old more interesting people--who know what the foundations of their thoughts actually assume. A classic.
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Product DescriptionVolume II, on formal (ZFC) set theory, incorporates a self-contained "chapter 0" on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques provides a solid foundation in set theory and a thorough context for the presentation of advanced topics (such as absoluteness, relative consistency results, two expositions of Godel's construstive universe, numerous ways of viewing recursion and Cohen forcing). ... Read more

Customer Reviews (1)

Excellent book as an advanced introduction.
My opinion is that this is the most readable and user friendly advanced book on Set Theory today. Especially the treatment of (Cohen's) Forcing is valuable, since other expositions on this subject were not suitable for beginning graduate students (Kunnen - very good, but very sophisticated,Jech - also good, but presents Forcing via Boolean Algebra, which is less widely used today, and the book is too long as an advance introduction). Also the style or writing is inviting, and not intimidating.Very significant book, pedagogically.
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Product DescriptionA succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics.Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. ... Read more

Customer Reviews (8)

Good
The book deals with the elementary parts of logic, computability and settheory from an algebraic and/or "abstract" point of view. Henceit is not really suitable as a first introduction to logic (except possiblyfor persons of extremely deep insight!) Of course nothing in the book isactually difficult. But the exposition is sketchy and lacks sufficientmotivation. Important foundational, motivational, historical side-topicsare ignored. The ideas and intuitions shaping the subject are relegated tothe background of slick technical developments. As I mentioned below, thesereally are just notes! Most novices ought to suffer a more traditionalexposure to logic first; such as reading [Enderton] or [Ebbinghaus et al.]

On the other hand, for people with *some* background and *some*mathematical inclination and *some* sense of mathematical beauty, this bookis fun. The abstract approach brings out the essential features of thenotions studied in logic, provides slick proofs and makes an implicit casefor the unity of mathematics including mathematical logic -- which is themathematical study of (various aspects of) mathematics itself. I personallylike these "abstractions" but if you don't like them or if youdon't yet have the necessary background, don't worry: There are other goodlogic books out there with a lighter touch.

Nice (algebraic) introduction
The book deals with the elementary parts of logic, computability and settheory from an algebraic and/or "abstract" point of view. Henceit is not really suitable as a first introduction to logic (except possiblyfor persons of extremely deep insight!) Of course nothing in the book isactually difficult. But the exposition is sketchy and lacks sufficientmotivation. Important foundational, motivational, historical side-topicsare ignored. The ideas and intuitions shaping the subject are relegated tothe background of slick technical developments. As I mentioned below, thesereally are just notes! Most novices ought to suffer a more traditionalexposure to logic first; such as reading [Enderton] or [Ebbinghaus et al.]

On the other hand, for people with *some* background and *some*mathematical inclination and *some* sense of mathematical beauty, this bookis fun. The abstract approach brings out the essential features of thenotions studied in logic, provides slick proofs and makes an implicit casefor the unity of mathematics including mathematical logic -- which is themathematical study of (various aspects of) mathematics itself. I personallylike these "abstractions" but if you don't like them or if youdon't yet have the necessary background, don't worry: There are other goodlogic books out there with a lighter touch.

interesting book
This is an interesting book.

Not very suitable for introduction.

But good nevertheless.

I esp. like the section on computability.

The logic and set theory were a bit too short.

I agree with the Vera Suslova thatthis is not for beginners!

Good
The book deals with the elementary parts of logic, computability and set theory from an algebraic and/or "abstract" point of view. Hence it is not really suitable as a first introduction to logic (except possiblyfor persons of extremely deep insight!) Of course nothing in the book isactually difficult. But the exposition is sketchy and lacks sufficientmotivation. Important foundational, motivational, historical side-topicsare ignored. The ideas and intuitions shaping the subject are relegated tothe background of slick technical developments. As I mentioned below, thesereally are just notes! Most novices ought to suffer a more traditionalexposure to logic first; such as reading [Enderton] or [Ebbinghaus et al.]

On the other hand, for people with *some* background and *some*mathematical inclination and *some* sense of mathematical beauty, this bookis fun. The abstract approach brings out the essential features of thenotions studied in logic, provides slick proofs and makes an implicit casefor the unity of mathematics including mathematical logic -- which is themathematical study of (various aspects of) mathematics itself. I personallylike these "abstractions" but if you don't like them or if youdon't yet have the necessary background, don't worry: There are other goodlogic books out there with a lighter touch.

Good
The book deals with the elementary parts of logic, computability and set theory from an algebraic and/or "abstract" point of view. Hence it is not really suitable as a first introduction to logic (except possiblyfor persons of extremely deep insight!) Of course nothing in the book isactually difficult. But the exposition is sketchy and lacks sufficientmotivation. Important foundational, motivational, historical side-topicsare ignored. The ideas and intuitions shaping the subject are relegated tothe background of slick technical developments. As I mentioned below, thesereally are just notes! Most novices ought to suffer a more traditionalexposure to logic first; such as reading [Enderton] or [Ebbinghaus et al.]

On the other hand, for people with *some* background and *some*mathematical inclination and *some* sense of mathematical beauty, this bookis fun. The abstract approach brings out the essential features of thenotions studied in logic, provides slick proofs and makes an implicit casefor the unity of mathematics including mathematical logic -- which is themathematical study of (various aspects of) mathematics itself. I personallylike these "abstractions" but if you don't like them or if youdon't yet have the necessary background, don't worry: There are other goodlogic books out there with a lighter touch.
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Product DescriptionThis two-volume work bridges the gap between introductory expositions of logic (or set theory) and the research literature.It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly lecture style that makes them equally effective for self-study or class use. Volume I includes formal proof techniques, applications of compactness (including nonstandard analysis), computability and its relation to the completeness phenonmenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen. ... Read more

Product DescriptionRather than teach mathematics and the structure of proofs simultaneously, this book first introduces logic as the foundation of proofs and then demonstrates how logic applies to mathematical topics. This method ensures that the reader gains a firm understanding of how logic interacts with mathematics and empowers them to solve more complex problems.Topics include: Propositional Logic; Predicates and Proofs; Set Theory; Mathematical Induction; Number Theory; Relations and Functions; Ring Theory; Topology. ... Read more

Customer Reviews (1)

A brilliant textbook by a brilliant Professor
Dr. O'Leary is a wonderful Professor, I had the pleasure of taken his class in Proof Theory that uses this text book. In that class we covered the first two sections of the book. We did not make it to the "Coming Attractions" section, however we were on a 12 week format perhaps the extra four weeks we could have made it.

The book is a reasonable read. Calculus isn't necessary only in the "You've been doing Math long enough" sort of benchmark.

This book provides a really fairly unique and at the very least distinct approach to foundations. Instead of the "Sink or Swim" method traditionally forced on students this text drives home the Symbolic Logic as the foundations (After all that is why Principia Mathematica [Russell Whitehead] was such a big deal last century right? Math = Logic) and the fundamental tools to for use in this thing we call Mathematics.

Anyone looking for a rigorous treatment of symbolic logic or Mathematical foundations via Logic and Set Theory this is the textbook for you. I found myself in both areas and I was pleasantly surprised with the quality of this textbook. I am thankful for both the book and the having had the chance to take the class using this book by the Author.
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Product DescriptionThis volume is both a tribute to Ulrich Felgner's research inalgebra, logic, and set theory and a strong researchcontribution to these areas. Felgner's former students, friendsand collaborators have contributed sixteen papers to thisvolume that highlight the unity of these three fields in the spiritof Ulrich Felgner's own research. The interested reader will find excellent original research surveys and papers that span the field from set theory without the axiom of choice via model-theoretic algebra to the mathematics of intonation. ... Read more

Product DescriptionProvides the reader with comprehensive coverage of theoretical foundations of fuzzy set theory and fuzzy logic, as well as a broad overview of the increasingly important applications of these novel areas of mathematics. DLC: Fuzzy sets. ... Read more

Customer Reviews (5)

Not just fuzzy
This book makes a parallel between regular math concepts and the ones that are used in the fuzzy logic. This was very evident to me when I was working with linear algebra, more precisely with linear programming. Nice book to have, even if this is only to know more about the subject than to really work with it.

Care is needed
I would hesistate to give anything less than a 5 star review to anything on fuzzy set theory in the wide sense.Make no mistake reading this book is worth your time.Yet, some significant problems do exist with this text.
First off, read the proofs in this carefully and figure out if they do work.Klir and Yuan know that appealing to contradiction in theorem proving doesn't often work out in fuzzy theory.Yet, they go ahead and use it almost recklessly.One example is their proof on fuzzy numbers that says that they are all continuous on pages 99 to 100.After about a full, condensed page of mathematical reasoning they say that left fuzzy numbers are continuous from the left and that right fuzzy numbers are continuous from the right.After their supposed "proof" they claim that "The implication of Theorem 4.1 is that every fuzzy number be represented in the form of (4.1)."4.1 shows a discontinuous fuzzy number.A jump discontinuity to speak more specifically.Consequently, their supposed "theorem" doesn't exactly work as a "theorem".Perhaps I misunderstand and they have some different idea of continuity.I don't get it though and neither does any other mathematician, as any break in a function whatsoever means discontinuity.
More interestingly, some of their axioms for fuzzy set don't hold.For instance, on page 62 Axiom i1 (i for intersection) says that i(a, 1)=a, which they label as the "boundary conidition."This does hold for drastic products.However, it doesn't hold for all fuzzy intersections.As Buckley and Eslami point out the axioms or necessary conditions for fuzzy intersections work out as "(1) 0<=a, b<=1 and i(a, b) is in
[0, 1]; (2) i(1, 1)=1; and (3) i(0, 1)=i(1, 0)=i(0, 0)=0."Consquently, (ab)/max{a, b, .5} qualifies as a fuzzy intersections.Here i(.6, .4)=.24/.6=24/60=2/5=.4
I don't exactly mean the above to significantly downgrade the work of Klir and Yuan.Their collection of papers of Zadeh does have signficant value, even if it costs a lot.The sheer enormity and very comprehensive nature of this quasi-encyclopedia makes it worth the read.The problems are interesting and challenging, if you choose to do them.I do appreciate the authors mentioning that the problems are meant to enchance the reader's understanding.That Klir and Yuan provide a comprehensive bibliography and consulted many, many original papers before and while writing their text alone indicates they do know something and did some thinking here.Their graphs do help to illustrate their ideas.So, I do advise that you read the book.Just read carefully.

First bible of fuzzy systems theory since Dubois and Prade.
A comprehensive and authoritative presentation of developments in themathematics of fuzzy systems theory over the past thiry years. While thebasic mathematics are presented, this book is not for the casual reader,but for those seriously interested in fuzzy systems theory. If the readerdoes not have a good mathematical background, he or she will find this booktough going. Coverage of theoretical fuzzy concepts is quite complete,including theory of fuzzy sets, fuzzy arithmetic, fuzzy relations,possiblity theory, fuzzy logic and uncertainty-based information.

Theapplications section presents theory which could be useful in applicationsrather than the applications themselves. References are given, but nodistinction is made between theoretical work and real-world applications,and many of the references are old and out-of-date.

For a reference bookon fuzzy mathematics, this book is superb; as a pointer to real-worldapplications, it leaves something to be desired.

Robust treatment of fuzzy logic has interdisciplinary appeal
George and Bo have been as thorough and lucid in preparing this book aswell as George explicated systems thinking in the very first book of his Iread, "An Approach to General Systems Theory." Here, as there,without compromising mathematical rigor, the goal of this book is toelaborate its subject matter in such a robust manner that it hasmultidisciplinary appeal. As always, the reader is given a flexible, almostinteractive, access to the what, why and how of fuzzy thinking. Despite theexception taken by Professor Lotfi A. Zadeh, the "founder of fuzzylogic," the percipient reader will appreciate the authors' unusualassociation of "fuzzy measure," that is, the degree of beliefthat a particular element belongs to a crisp set, (not the degree ofmembership in the set), with Possibility Theory so as to clarify thedifferences between fuzzy set theory and probability theory. Theillustrative applications are not only case studies that one may pick andchoose from for examination and emulation but also constituteincontrovertible evidence of the successful and promising realization ofthe fuzzy paradigm. As a former professor of engineering at RutgersUniversity, I found the 79-page Instructor's manual helpful for self- orextended study and Iassume it would be valuable for teaching. I have readmany books on fuzzy logic and I judge this to be the most balanced to date,(early 1998), - not filled with C++ code or trying to sell a softwarepackage nor is it theoretically daunting - it is simply an invitingdemonstration of how fuzzy logic clears up foggy modeling and analysis.

One of the most important book to learn about fuzzy logic
The book presents the mathematical theory of fuzzy logic including theorems and demonstrations. There are one part of applications of this logic in many distint areas like engineering, medicine, economics and others.
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Product DescriptionWillard Van Orman Quine (1908-2000) was probably the most influential American philosopher of the twentieth century. In Trading Ontology for Ideology Lieven Decock offers an insightful analysis of the development of Quine's ontological views from his first texts in the early thirties onwards. The importance of Quine's work in logic and set theory for his ontology is highlighted. Decock argues that the tenet of extensionalism is at least as important as naturalism, and assesses the relation between the two. The other focus of the work is the relation between ontology, i.e. what there is, and ideology, i.e. what can be expressed by means of words. Decock shows that the interplay between ontology and ideology is far more complicated and interesting than has generally been assumed. ... Read more

THE book for studying Boolean-valued models of set theory
If you are studying to become a set theorist these days (2001 or later), you will probably learn forcing by way of partial orders. This is the quickest way to go, and for most purposes, the most useful in applications. It's also the approach that is taken in Kunen's book on set theory, which is still pretty much the standard textbook.

Nonetheless, you may one day find that you need the Boolean algebra approach. This was the approach developed by Scott and Solovay after Cohen's somewhat inscrutable approach to proving the consistency of not-CH had been adequately digested. The Boolean algebra approach isvery elegant and algebraic, and the theorems are often bettermotivated than they are in the partial orders approach.

When I was a graduate student, I studied both approaches,but eventually forgot much of what I learned about Boolean-valued models because in practice I always relied on the other approach.

Recently, however, in some research I've been doing in an extension of ZFC in an extended language, I found that some of the usual assumptions one can make in doing forcing over ZFC models were no longer applicable in the new setting. Without going into the details, the consequence was that I had to do forcing over non-wellfounded models of ZFC and examine the properties of the resulting forcing extension, perhaps iterating the process omega many times. The only way to do this has been to work with Boolean-valued models M^B of the non-wellfounded ground model M, prove the desired properties within M^B, then collapse with a generic ultrafilter, and go on to the next model.

Well, after that long-winded introduction, my point is this: Bell's treatment of Boolean-valued models is outstanding. I have several of Bell's books and his talent as an expositor is his relentless attention to detail. He does no hand-waving. If you need to face the details of Boolean-valued models, Bell's approach is the right way to go.

In the first chapter he develops the theory enough to prove that all ZFC axioms hold in V^B. In Chapter 2 he shows how to do independence proofs in Boolean valued models -- illustrating with CH and developing the usual results about chain conditions and distributivity, never once working with a 2-valued forcing extension. The third chapter reveals some of the elegance of the Boolean algebra approach in its development of the proof of the consistency of not-AC using group actions. Chapter 4 shows how to get the usual results about forcing involving 2-valued models by considering (V^B)/U, where U is a generic ultrafilter. Chapter 5 is a special chapter about cardinal collapsing, introduced because the Boolean algebras introduced before always preserved cardinals and cofinalities. Finally, chapter 6, which was added in the 1985 edition, treats iterated forcing. This chapter contains details that appear nowhere else and are very handy if you need to deal with such things.

As ever, Bell has done a thorough job in his treatment of this subject. It is the right reference for Boolean-valued models of set theory.
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'A Geometry of Approximation' addresses Rough Set Theory, a field of interdisciplinary research first proposed by Zdzislaw Pawlak in 1982, and focuses mainly on its logic-algebraic interpretation. The theory is embedded in a broader perspective that includes logical and mathematical methodologies pertaining to the theory, as well as related epistemological issues. Any mathematical technique that is introduced in the book is preceded by logical and epistemological explanations. Intuitive justifications are also provided, insofar as possible, so that the general perspective is not lost.

Such an approach endows the present treatise with a unique character. Due to this uniqueness in the treatment of the subject, the book will be useful to researchers, graduate and pre-graduate students from various disciplines, such as computer science, mathematics and philosophy. It features an impressive number of examples supported by about 40 tables and 230 figures. The comprehensive index of concepts turns the book into a sort of encyclopaedia for researchers from a number of fields.

'A Geometry of Approximation' links many areas of academic pursuit without losing track of its focal point, Rough Sets.

Product DescriptionWritten for professionals learning the field of discrete mathematics, this book provides the necessary foundations of computer science without requiring excessive mathematical prerequisites. Using a balanced approach of theory and examples, software engineers will find it a refreshing treatment of applications in programming. ... Read more

Product DescriptionProblems in Set Theory, Mathematical Logic and the Theory of Algorithms by I. Lavrov & L. Maksimova is an English translation of the fourth edition of the most popular student problem book in mathematical logic in Russian. It covers major classical topics in proof theory and the semantics of propositional and predicate logic as well as set theory and computation theory. Each chapter begins with 1-2 pages of terminology and definitions that make the book self-contained. Solutions are provided. The book is likely to become an essential part of curricula in logic. ... Read more